WASHINGTON eXPERIMENTAL MATHEMATICS LAB

# Winter 2018

## Logistics:

WXML general meetings will be from 5-6pm on the following three Thursdays in Mary Gates Hall (MGH) room 295:

• First meeting: January 4
• Mid-quarter meeting: February 1
• Final meeting: March 1

The WXML Winter 2018 Open House will be on March 9, from 5-7pm, in Odegaard Active Learning Classroom 136. (Note: this is different from the original schedule.)

WXML has reserved MGH 295 on Tuesdays and Thursdays from 5-7pm for all of Winter quarter for groups to use as a meeting space. The WXML lab in the basement of the Communications building is also available.

## Projects and Teams:

### Algebraic Combinatorics

Faculty Mentors: Sara Billey and Philippe Nadeau (Math)

Team Members: Justin Shyi, Jesse Rivera

In Algebra, a familiar object is the set of cosets G/H for a given group G and subgroup H. There are also many reasons to study double cosets H\G/K. If G is the symmetric group S_n and H and K are Young subgroups, then the double cosets are closely related to representation theory and also to the Bruhat order. In Combinatorics, we are interested in studying efficient formulas for counting discrete objects. Double cosets of symmetric groups are discrete objects, so they deserved to be counted! But, what is the best possible formula? What are the interesting types of subgroups to consider? This work has connections to prior research by the mentors on tanglegrams, parabolic double cosets, and pattern-avoiding permutations.

### Orbit structure of crystal operators

Faculty Mentor: Jake Levinson (Math)

Team Members: Yujin Jeong, Junchen Pan

We’re going to study a transformation (algorithm) involving a type of diagram called a Young tableau. The transformation is combinatorial, applying a sequence of ‘moves’ to the diagram. The algorithm’s significance comes from geometry, where it describes the result of varying a parameter (i.e. moving a point) along a special kind of curve. After iterating the algorithm enough times, the diagram will be transformed back to its original form, which indicates that the point has traveled around a loop, back to where it started. I’d like to understand the orbit structure of this transformation. That is, given a diagram, how many steps does it take for it to return to its original form? And, given a set of diagrams (= points on the curve), how many distinct orbits do they form (= number of loops formed by the curve)? Can we tell if two diagrams are in the same orbit? Goals of the project. Code up the algorithm and run it, collecting data about the orbit structure of the transformation. Test out some conjectures about identifying and counting orbits. And, if there’s interest, learn about some of the geometry as well — where this combinatorics comes from.

[Final report]

### Mathematics of Gerrymandering

Faculty Mentor: Christopher Hoffman (Math)

Team Members: Leo Segovia, Alexander Robkin, Weifan Jiang, Namyoung Kim

Districting and Gerrymandering have been in the news a lot recently, and the mathematical modeling of how to draw districts is a very hot topic. This project will look at the space of all possible redistrictings of a state to see whether the current plan is an outlier. It will involve probability and computing skills.

[Final report]

### Counting k-tuples in discrete sets

Team Members: Andrew Lim, Kimberly Bautista, Madeline Brown

We will count pairs, triples, and k-tuples of integer vectors in the plane, 3-space, and higher dimensions, using both number theory and programming. We’ll also explore other interesting sets arising from dynamical systems and billiards. This project will involve number theory, geometry, and probability, and will be a great introduction to those subjects.

### Rotation Random Walks

Project Mentors: Jayadev Athreya and Chris Hoffman (Math)

Team Members: Peter Gylys-Colwell, Alex Tsun, Katrina Warner

We’ll study a simple random walk on the circle given by rotating clockwise or counterclockwise by a fixed angle, based on flipping a fair coin. We’ll try and understand the fine distribution of this walk, using both numerical experiments and theory.

### Uniformity of solutions of Diophantine Equations

Project Mentor: Amos Turchet (Math)

Team Members: Bryan Quah, Blanca Viña Patiño, Rohan Hiatt, Daria Micovic

Given an equation in two variables with integer coefficients, how many solutions can you find where both values of the variables are rational numbers? This apparently easy question turned out to be very deep and very interesting from the point of view of Number Theory and Algebraic Geometry. In this project we will investigate the behavior of such solutions set for special equations representing so called, higher genus curves. We will gather some statistic on the size of the solutions set trying to address an important open problem on the uniformity of these sizes.

### Number Theory and Noise

Project Mentor: Matthew Conroy (Math)

Graduate Mentors: Nikolas Eptaminitakis and Gabriel Dorfsman-Hopkins

Team Members: Nile Wynar, Xiaotong Chen, Robert Pedersen, Hongyu Cao

The project will investigate integer sequences via sound. This quarter, students will add to the library of integer sequence audio files (the expectation will be that each student will add 5 to 10 new audio files each week). Details and many examples are here: http://www.math.washington.edu/~conroy/sequenceNoise/index.htm

### Conditional path sampling of metastable states

Project Mentor: Panos Stinis (AMath)